Allocating Railway Traffic with QUBO Formulated Models
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Who we are
Ferrovie dello Stato Italiane S.p.A
• more than 10000 trains per day;
• 16700km long network;
• more then 2000 stations;
• 700 million passengers per year.
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
27/03/2019
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Who we are
Data Science and Digital Transformation
• Experiments in Innovation:
• AI and Machine Learning
• IoT
• Blockchain
• Quantum Technologies
• ...
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Optimization problems at FS
Many hard optimization problems:
• Workforce deployment;• deployment of trains for maintenance;• multimodal optimization of time table schedule;• load optimization for freight trains;• ...
Some of these are currently optimized by automatic or semi-automatic engines while other aremanually solved.
Can we use Quantum Technologies to solve some of these?
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Train Platforming Problem (TPP)
• Operational Contraints:
• 1-to-1 assignment of train to platform (nooverlapping, no double assignment)
• Station topology (not every track leads to everydirection)
The problem of assigning a platform to each train approaching a station given a predefined timetable.
• Commercial Optimization
• Eg. Distanze walked by users inside the stationwhen switchin trains
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Solving the TPP in every day planning
Currently the TPP in FS is solved:
• semi manually
• according to a small set of heuristics
• with no commercial optimization
• In order to introduce an objective function the problem mustbe transformed into a quadratic constrained optimization
• The formulation of the Train Platforming Problem (TPP) is:
• Next approach: QUBO
→ faster to solve and suitable to be run on a quantum computer.
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Quadratic model!
(*) Ref: Solution of the train platforming problem. https://doi.org/10.1287/trsc.1100.0366
(*)
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Business interest
The model:
• it facilitates the flow of passengers within the station
• allows a better planning of the station workforce deployment as well as resources logistic
• Increases station capacity
• Could also be applied to multimodal passenger connectivity optimizing the mobility through multipletransportation systems (train, bus, tram, etc).
Know how:
• Apply the know-how to other optimizations problem
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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The dataset
• Firenze Santa Maria Novella
• Platforms: 19
• Trains per day: ~400
Regional and long haul (also busses...)
• Simple topology: all the platforms are on the same level
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Train Traffic Analysis
• More than 2000 temporal overlapping to avoid
• Total number of variables is about 7000
• Rescheduling in near real time requires huge computational power
• Considering the effects of a delay on the planning is challenging
• Optimizing the station as part of the network (propagation of delays and maximization of the network effect)
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3:00 9:00 15:00 21:00
Time
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Dwave computational model
Problems can be formulated as:
1. Ising models:
2. QUBO optimization:
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𝐻𝑖𝑠𝑖𝑛𝑔 =
𝑖=0
𝑁
ℎ𝑖𝑠𝑖 +
𝑗>𝑖
𝐽𝑖𝑗𝑠𝑖 𝑠𝑗
min𝑥
𝑗≥𝑖
𝑥𝑖 𝑄𝑖𝑗𝑥𝑗
A particular mapping has to be used to elaborate data through quantum annealing
• Q is the matrix that encodes our problem
• xi are binary variables representing pairs (train, platform)
• xi = 1 if corresponding train t is placed on platform p
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Processing data with Quantum Annealing
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QUBO objective function
1. Data analysis and preparation
2. QUBO formulation of the problem
3. Solution through classical Qbsolv or Dwave quantum annealing
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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QUBO objective function
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The Hamiltonian to be minimized can be divided into:
The penalties term modelling the optimization of the station
𝐸 𝑥
The term related to the single assignment of a train
The part responsible for the temporal overlapping of different trains on the same platform
Operational constraints
Optimization part
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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The optimization
• Optimization measure: Walking distance toswitch trains
• We consider trains to be in coincidence onlywithin 1 hour of temporal window.
• Allow train with longer coincidence time to beplaced further apart
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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The distance matrix
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Evaluating the distances of platforms
from the first and 5th platform:
0 5 10 15
Platforms
0
5
10
15
Pla
tfo
rms
Me
ters
015
30
45
60
75
90
105
120
Platform 5
Platform 0
Platform
2 4 6 8 10 12 14 16
Me
ters
20
40
60
80
100
120
Modelling the walking distance between different set of platforms:
By using this calibration it is possible to calculate the QUBO configuration of the problem
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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The QUBO matrix
• The QUBO matrix encapsulates all the data and constraints
• All the platforms are initially considered as equally important: this is a simplification
• Additional constraints consider the connectivity between platforms
• Huge dataset: this QUBO contains about 7000 variables
We solved this matrix by using the Dwave Qbsolv running a classical tabu-search on a HPC cluster
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Feasible solution:
• No overlapping
• No optimization
Solution
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Optimized solution:
• No overlapping
• Optimized
Solution
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Quality: 30% better placement (measured by average distance walked)
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Computational time
All our experimentations on real datasets so far have beenperformed on classical hardware
On a single node laptop, the problem was solved in about 15minutes
On a HPC cluster this QUBO formulation was solved in about 2minutes; considerably fast, but still not enough to be usable inreal time
The next step would be to test the solver on a real QPU
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Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
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Future perspectives.
• Test the algorithm on a quantum device
• Develop further know-how on Quantumapproaches to optimization
Publication and Collaboration.
• Work accepted at World Railway Research Conference (Tokyo, Oct 2019)
Conclusion and Future Perspectives
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Further Improvements.
• Generalize to different station layouts
• Distance matrix changes, but overall algorithm is exactly the same
• Consider platforms connectivity:
• Not all the platforms are connected with specific directions
• Consider multimodality:
• Bus terminal near the station, parking lot for car sharing services...
• Take into account real time changes:
• How to reschedule the platforming in case of delays and/or disruptive changes (e.g.: unusable platforms, maintenance, …)?
Allocating Railway Traffic with QUBO Formulated Models | Lorenzo Ferrone,Davide Caputo
Questions?